# On a Bernoulli problem with geometric constraints

Antoine Laurain; Yannick Privat

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 1, page 157-180
- ISSN: 1292-8119

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topLaurain, Antoine, and Privat, Yannick. "On a Bernoulli problem with geometric constraints." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 157-180. <http://eudml.org/doc/221908>.

@article{Laurain2012,

abstract = {A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane \{x1 = 0\} where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints. },

author = {Laurain, Antoine, Privat, Yannick},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Free boundary problem; Bernoulli condition; shape optimization; free boundary problem},

language = {eng},

month = {2},

number = {1},

pages = {157-180},

publisher = {EDP Sciences},

title = {On a Bernoulli problem with geometric constraints},

url = {http://eudml.org/doc/221908},

volume = {18},

year = {2012},

}

TY - JOUR

AU - Laurain, Antoine

AU - Privat, Yannick

TI - On a Bernoulli problem with geometric constraints

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2012/2//

PB - EDP Sciences

VL - 18

IS - 1

SP - 157

EP - 180

AB - A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane {x1 = 0} where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

LA - eng

KW - Free boundary problem; Bernoulli condition; shape optimization; free boundary problem

UR - http://eudml.org/doc/221908

ER -

## References

top- A. Acker, An extremal problem involving current flow through distributed resistance. SIAM J. Math. Anal.12 (1981) 169–172.
- C. Atkinson and C.R. Champion, Some boundary-value problems for the equation ∇·(|∇ϕ| N∇ϕ) = 0. Quart. J. Mech. Appl. Math.37 (1984) 401–419.
- A. Beurling, On free boundary problems for the Laplace equation, Seminars on analytic functions1. Institute for Advanced Studies, Princeton (1957).
- F. Bouchon, S. Clain and R. Touzani, Numerical solution of the free boundary Bernoulli problem using a level set formulation. Comput. Methods Appl. Mech. Eng.194 (2005) 3934–3948.
- E.N. Dancer and D. Daners, Domain perturbation for elliptic equations subject to Robin boundary conditions. J. Differ. Equ.138 (1997) 86–132.
- M.C. Delfour and J.-P. Zolésio, Shapes and geometries – Analysis, differential calculus, and optimization, Advances in Design and Control4. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2001).
- L.C. Evans, Partial differential equations, Graduate Studies in Mathematics19. American Mathematical Society, Providence (1998).
- A. Fasano, Some free boundary problems with industrial applications, in Shape optimization and free boundaries (Montreal, PQ, 1990), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.380, Kluwer Acad. Publ., Dordrecht (1992) 113–142.
- M. Flucher and M. Rumpf, Bernoulli’s free boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math.486 (1997) 165–204.
- A. Friedman, Free boundary problem in fluid dynamics, in Variational methods for equilibrium problems of fluids, Trento 1983, Astérisque118 (1984) 55–67.
- A. Friedman, Free boundary problems in science and technology. Notices Amer. Math. Soc.47 (2000) 854–861.
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics24. Pitman (Advanced Publishing Program), Boston (1985).
- J. Haslinger, T. Kozubek, K. Kunisch and G. Peichl, Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. Comp. Optim. Appl.26 (2003) 231–251.
- J. Haslinger, K. Ito, T. Kozubek, K. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type. Interfaces in Free Boundaries11 (2009) 317–330.
- A. Henrot and M. Pierre, Variation et optimisation de formes – Une analyse géométrique, Mathématiques & Applications48. Springer, Berlin (2005).
- A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the p-Laplace operator. I. The exterior convex case. J. Reine Angew. Math.521 (2000) 85–97.
- A. Henrot and H. Shahgholian, Existence of classical solutions to a free boundary problem for the p-Laplace operator. II. The interior convex case. Indiana Univ. Math. J.49 (2000) 311–323.
- A. Henrot and H. Shahgholian, The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition. Trans. Amer. Math. Soc.354 (2002) 2399–2416.
- K. Ito, K. Kunisch and G.H. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems. J. Math. Anal. Appl.314 (2006) 126–149.
- C.T. Kelley, Iterative methods for optimization, Frontiers in Applied Mathematics18. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999).
- V.A. Kondratév, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč.16 (1967) 209–292.
- C.M. Kuster, P.A. Gremaud and R. Touzani, Fast numerical methods for Bernoulli free boundary problems. SIAM J. Sci. Comput.29 (2007) 622–634.
- J. Lamboley and A. Novruzi, Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim.48 (2009) 3003–3025.
- E. Lindgren and Y. Privat, A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition. Nonlinear Anal.67 (2007) 2497–2505.
- J. Nocedal and S.J. Wright, Numerical optimization. Springer Series in Operations Research and Financial Engineering, Springer, New York, 2nd edition (2006).
- J.R. Philip, n-diffusion. Austral. J. Phys.14 (1961) 1–13.
- J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization : Shape sensitivity analysis, Springer Series in Computational Mathematics16. Springer-Verlag, Berlin (1992).

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